# Properties

 Label 1800.1.ba.b Level $1800$ Weight $1$ Character orbit 1800.ba Analytic conductor $0.898$ Analytic rank $0$ Dimension $4$ Projective image $D_{3}$ CM discriminant -8 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,1,Mod(499,1800)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 2, 3]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1800.499");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1800 = 2^{3} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1800.ba (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$0.898317022739$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 72) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.648.1 Artin image: $S_3\times C_{12}$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{24} - \cdots)$$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - \zeta_{12}^{5} q^{2} + \zeta_{12}^{5} q^{3} - \zeta_{12}^{4} q^{4} + \zeta_{12}^{4} q^{6} - \zeta_{12}^{3} q^{8} - \zeta_{12}^{4} q^{9} +O(q^{10})$$ q - z^5 * q^2 + z^5 * q^3 - z^4 * q^4 + z^4 * q^6 - z^3 * q^8 - z^4 * q^9 $$q - \zeta_{12}^{5} q^{2} + \zeta_{12}^{5} q^{3} - \zeta_{12}^{4} q^{4} + \zeta_{12}^{4} q^{6} - \zeta_{12}^{3} q^{8} - \zeta_{12}^{4} q^{9} + \zeta_{12}^{2} q^{11} + \zeta_{12}^{3} q^{12} - \zeta_{12}^{2} q^{16} - \zeta_{12}^{3} q^{17} - \zeta_{12}^{3} q^{18} + q^{19} + \zeta_{12} q^{22} + \zeta_{12}^{2} q^{24} + \zeta_{12}^{3} q^{27} - \zeta_{12} q^{32} - \zeta_{12} q^{33} - \zeta_{12}^{2} q^{34} - \zeta_{12}^{2} q^{36} - \zeta_{12}^{5} q^{38} - \zeta_{12}^{4} q^{41} - \zeta_{12}^{5} q^{43} + q^{44} + \zeta_{12} q^{48} - \zeta_{12}^{4} q^{49} + \zeta_{12}^{2} q^{51} + \zeta_{12}^{2} q^{54} + \zeta_{12}^{5} q^{57} + \zeta_{12}^{4} q^{59} - q^{64} - q^{66} + \zeta_{12} q^{67} - \zeta_{12} q^{68} - \zeta_{12} q^{72} + \zeta_{12}^{3} q^{73} - \zeta_{12}^{4} q^{76} - \zeta_{12}^{2} q^{81} - \zeta_{12}^{3} q^{82} + \zeta_{12}^{5} q^{83} - \zeta_{12}^{4} q^{86} - \zeta_{12}^{5} q^{88} - q^{89} + q^{96} + \zeta_{12}^{5} q^{97} - \zeta_{12}^{3} q^{98} + q^{99} +O(q^{100})$$ q - z^5 * q^2 + z^5 * q^3 - z^4 * q^4 + z^4 * q^6 - z^3 * q^8 - z^4 * q^9 + z^2 * q^11 + z^3 * q^12 - z^2 * q^16 - z^3 * q^17 - z^3 * q^18 + q^19 + z * q^22 + z^2 * q^24 + z^3 * q^27 - z * q^32 - z * q^33 - z^2 * q^34 - z^2 * q^36 - z^5 * q^38 - z^4 * q^41 - z^5 * q^43 + q^44 + z * q^48 - z^4 * q^49 + z^2 * q^51 + z^2 * q^54 + z^5 * q^57 + z^4 * q^59 - q^64 - q^66 + z * q^67 - z * q^68 - z * q^72 + z^3 * q^73 - z^4 * q^76 - z^2 * q^81 - z^3 * q^82 + z^5 * q^83 - z^4 * q^86 - z^5 * q^88 - q^89 + q^96 + z^5 * q^97 - z^3 * q^98 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} - 2 q^{6} + 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^4 - 2 * q^6 + 2 * q^9 $$4 q + 2 q^{4} - 2 q^{6} + 2 q^{9} + 2 q^{11} - 2 q^{16} + 4 q^{19} + 2 q^{24} - 2 q^{34} - 2 q^{36} + 2 q^{41} + 4 q^{44} + 2 q^{49} + 2 q^{51} + 2 q^{54} - 2 q^{59} - 4 q^{64} - 4 q^{66} + 2 q^{76} - 2 q^{81} + 2 q^{86} - 8 q^{89} + 4 q^{96} + 4 q^{99}+O(q^{100})$$ 4 * q + 2 * q^4 - 2 * q^6 + 2 * q^9 + 2 * q^11 - 2 * q^16 + 4 * q^19 + 2 * q^24 - 2 * q^34 - 2 * q^36 + 2 * q^41 + 4 * q^44 + 2 * q^49 + 2 * q^51 + 2 * q^54 - 2 * q^59 - 4 * q^64 - 4 * q^66 + 2 * q^76 - 2 * q^81 + 2 * q^86 - 8 * q^89 + 4 * q^96 + 4 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1001$$ $$1351$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\zeta_{12}^{4}$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
499.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.866025 + 0.500000i 0.866025 0.500000i 0.500000 0.866025i 0 −0.500000 + 0.866025i 0 1.00000i 0.500000 0.866025i 0
499.2 0.866025 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i 0 −0.500000 + 0.866025i 0 1.00000i 0.500000 0.866025i 0
1699.1 −0.866025 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 0 −0.500000 0.866025i 0 1.00000i 0.500000 + 0.866025i 0
1699.2 0.866025 + 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i 0 −0.500000 0.866025i 0 1.00000i 0.500000 + 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
5.b even 2 1 inner
9.c even 3 1 inner
40.e odd 2 1 inner
45.j even 6 1 inner
72.p odd 6 1 inner
360.z odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1800.1.ba.b 4
5.b even 2 1 inner 1800.1.ba.b 4
5.c odd 4 1 72.1.p.a 2
5.c odd 4 1 1800.1.bk.d 2
8.d odd 2 1 CM 1800.1.ba.b 4
9.c even 3 1 inner 1800.1.ba.b 4
15.e even 4 1 216.1.p.a 2
20.e even 4 1 288.1.t.a 2
35.f even 4 1 3528.1.cg.a 2
35.k even 12 1 3528.1.ba.a 2
35.k even 12 1 3528.1.ce.b 2
35.l odd 12 1 3528.1.ba.b 2
35.l odd 12 1 3528.1.ce.a 2
40.e odd 2 1 inner 1800.1.ba.b 4
40.i odd 4 1 288.1.t.a 2
40.k even 4 1 72.1.p.a 2
40.k even 4 1 1800.1.bk.d 2
45.j even 6 1 inner 1800.1.ba.b 4
45.k odd 12 1 72.1.p.a 2
45.k odd 12 1 648.1.b.b 1
45.k odd 12 1 1800.1.bk.d 2
45.l even 12 1 216.1.p.a 2
45.l even 12 1 648.1.b.a 1
60.l odd 4 1 864.1.t.a 2
72.p odd 6 1 inner 1800.1.ba.b 4
80.i odd 4 1 2304.1.o.c 4
80.j even 4 1 2304.1.o.c 4
80.s even 4 1 2304.1.o.c 4
80.t odd 4 1 2304.1.o.c 4
120.q odd 4 1 216.1.p.a 2
120.w even 4 1 864.1.t.a 2
180.v odd 12 1 864.1.t.a 2
180.v odd 12 1 2592.1.b.a 1
180.x even 12 1 288.1.t.a 2
180.x even 12 1 2592.1.b.b 1
280.y odd 4 1 3528.1.cg.a 2
280.bp odd 12 1 3528.1.ba.a 2
280.bp odd 12 1 3528.1.ce.b 2
280.br even 12 1 3528.1.ba.b 2
280.br even 12 1 3528.1.ce.a 2
315.bs even 12 1 3528.1.ba.a 2
315.bt odd 12 1 3528.1.ba.b 2
315.cb even 12 1 3528.1.cg.a 2
315.cg even 12 1 3528.1.ce.b 2
315.ch odd 12 1 3528.1.ce.a 2
360.z odd 6 1 inner 1800.1.ba.b 4
360.bo even 12 1 72.1.p.a 2
360.bo even 12 1 648.1.b.b 1
360.bo even 12 1 1800.1.bk.d 2
360.br even 12 1 864.1.t.a 2
360.br even 12 1 2592.1.b.a 1
360.bt odd 12 1 216.1.p.a 2
360.bt odd 12 1 648.1.b.a 1
360.bu odd 12 1 288.1.t.a 2
360.bu odd 12 1 2592.1.b.b 1
720.cn odd 12 1 2304.1.o.c 4
720.cp even 12 1 2304.1.o.c 4
720.cr odd 12 1 2304.1.o.c 4
720.ct even 12 1 2304.1.o.c 4
2520.hj odd 12 1 3528.1.ba.a 2
2520.hq even 12 1 3528.1.ba.b 2
2520.hw odd 12 1 3528.1.cg.a 2
2520.iy odd 12 1 3528.1.ce.b 2
2520.jd even 12 1 3528.1.ce.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.1.p.a 2 5.c odd 4 1
72.1.p.a 2 40.k even 4 1
72.1.p.a 2 45.k odd 12 1
72.1.p.a 2 360.bo even 12 1
216.1.p.a 2 15.e even 4 1
216.1.p.a 2 45.l even 12 1
216.1.p.a 2 120.q odd 4 1
216.1.p.a 2 360.bt odd 12 1
288.1.t.a 2 20.e even 4 1
288.1.t.a 2 40.i odd 4 1
288.1.t.a 2 180.x even 12 1
288.1.t.a 2 360.bu odd 12 1
648.1.b.a 1 45.l even 12 1
648.1.b.a 1 360.bt odd 12 1
648.1.b.b 1 45.k odd 12 1
648.1.b.b 1 360.bo even 12 1
864.1.t.a 2 60.l odd 4 1
864.1.t.a 2 120.w even 4 1
864.1.t.a 2 180.v odd 12 1
864.1.t.a 2 360.br even 12 1
1800.1.ba.b 4 1.a even 1 1 trivial
1800.1.ba.b 4 5.b even 2 1 inner
1800.1.ba.b 4 8.d odd 2 1 CM
1800.1.ba.b 4 9.c even 3 1 inner
1800.1.ba.b 4 40.e odd 2 1 inner
1800.1.ba.b 4 45.j even 6 1 inner
1800.1.ba.b 4 72.p odd 6 1 inner
1800.1.ba.b 4 360.z odd 6 1 inner
1800.1.bk.d 2 5.c odd 4 1
1800.1.bk.d 2 40.k even 4 1
1800.1.bk.d 2 45.k odd 12 1
1800.1.bk.d 2 360.bo even 12 1
2304.1.o.c 4 80.i odd 4 1
2304.1.o.c 4 80.j even 4 1
2304.1.o.c 4 80.s even 4 1
2304.1.o.c 4 80.t odd 4 1
2304.1.o.c 4 720.cn odd 12 1
2304.1.o.c 4 720.cp even 12 1
2304.1.o.c 4 720.cr odd 12 1
2304.1.o.c 4 720.ct even 12 1
2592.1.b.a 1 180.v odd 12 1
2592.1.b.a 1 360.br even 12 1
2592.1.b.b 1 180.x even 12 1
2592.1.b.b 1 360.bu odd 12 1
3528.1.ba.a 2 35.k even 12 1
3528.1.ba.a 2 280.bp odd 12 1
3528.1.ba.a 2 315.bs even 12 1
3528.1.ba.a 2 2520.hj odd 12 1
3528.1.ba.b 2 35.l odd 12 1
3528.1.ba.b 2 280.br even 12 1
3528.1.ba.b 2 315.bt odd 12 1
3528.1.ba.b 2 2520.hq even 12 1
3528.1.ce.a 2 35.l odd 12 1
3528.1.ce.a 2 280.br even 12 1
3528.1.ce.a 2 315.ch odd 12 1
3528.1.ce.a 2 2520.jd even 12 1
3528.1.ce.b 2 35.k even 12 1
3528.1.ce.b 2 280.bp odd 12 1
3528.1.ce.b 2 315.cg even 12 1
3528.1.ce.b 2 2520.iy odd 12 1
3528.1.cg.a 2 35.f even 4 1
3528.1.cg.a 2 280.y odd 4 1
3528.1.cg.a 2 315.cb even 12 1
3528.1.cg.a 2 2520.hw odd 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1800, [\chi])$$:

 $$T_{11}^{2} - T_{11} + 1$$ T11^2 - T11 + 1 $$T_{19} - 1$$ T19 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$T^{4} - T^{2} + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - T + 1)^{2}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 1)^{2}$$
$19$ $$(T - 1)^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$(T^{2} - T + 1)^{2}$$
$43$ $$T^{4} - T^{2} + 1$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$(T^{2} + T + 1)^{2}$$
$61$ $$T^{4}$$
$67$ $$T^{4} - T^{2} + 1$$
$71$ $$T^{4}$$
$73$ $$(T^{2} + 1)^{2}$$
$79$ $$T^{4}$$
$83$ $$T^{4} - 4T^{2} + 16$$
$89$ $$(T + 2)^{4}$$
$97$ $$T^{4} - T^{2} + 1$$